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302                      ANALYTICAL MECHANICS
a as the limits of x and the corresponding values of * as the limits of t we have
?                                   -r,         1
f                       =
4                  P                                        2 a
4 27r2a2m Cl . 227r, ,.
1.   A particle which describes a simple harmonic motion has a period of 5 sec. and an amplitude of 30 cm.   Find its maximum velocity and its maximum acceleration.
2.   When a load of mass m is suspended from a helical spring of length L and of negligible mass an extension equal to D is produced.   The load is pulled down through a distance a from its position of equilibrium and then set free.   Find the period and the amplitude of the vibration.    Hooke/s law holds true.
3.   Within the earth the gravitational attraction varies as the distance from the center.   Suppose there were a straight shaft from pole to pole, with no resisting medium in it.   What would be the period of oscillation of a body dropped into the shaft?   Suppose the earth to be a sphere with a radius of 4000 miles.
4.   In the preceding problem find the velocity with which the body would pass the center of the earth.
6. A particle describes a circle with constant speed. Show that the projection of the particle upon a straight line describes a simple harmonic motion.
6. The pan of a helical spring balance is lowered 2 inches when a weight of 5 pounds is placed on it. Find the period of vibration of the balance with the weight on.
* See footnote p. 142.